Using AI to further debunk ancient Egyptians used technologies to drill granite far beyond the current level.

[sjastro]

I was just reminiscing of the olds days before I discovered the one true computer language.

Don't tell me FORTRAN.

 

[SelfSim]

Don't tell me FORTRAN.

Ughh .. (don't remind me)! .. Batch queues using a card reader only to find out days later a "." was overlooked!
Progressed to Pascal, in my case.

 

[sjastro]

Ughh .. (don't remind me)! .. Batch queues using a card reader only to find out days later a "." was overlooked!
Progressed to Pascal, in my case.

The only time I ever used FORTRAN at work was calculating the repeatability and reproducibility for a test developed to measure the deterioration of rubber cups in brake master cylinders when brake fluid absorbed moisture.
This could lead to catastrophic brake failure through loss of brake fluid and a test was developed and then evaluated by various laboratories around the globe.

Since this thread involves AI here is a diagram of a brake master cylinder with the brake cup labelled as 'Seal' next to the return spring.

I asked GPT-4o to come with up with a diagram highlighting the cup in a brake master cylinder and it came up with this contraption.

​

Since references to Star Trek have been made in this thread, maybe this device has some relevance, I'm struggling with terms like 'Briston' and 'cymder' as well as the faded text along with the brake cups which pop up all over the place.

​

 

[SelfSim]

The only time I ever used FORTRAN at work was calculating the repeatability and reproducibility for a test developed to measure the deterioration of rubber cups in brake master cylinders when brake fluid absorbed moisture.
This could lead to catastrophic brake failure through loss of brake fluid and a test was developed and then evaluated by various laboratories around the globe.

Since this thread involves AI here is a diagram of a brake master cylinder with the brake cup labelled as 'Seal' next to the return spring.

I asked GPT-4o to come with up with a diagram highlighting the cup in a brake master cylinder and it came up with this contraption.

View attachment 358826
​

Since references to Star Trek have been made in this thread, maybe this device has some relevance, I'm struggling with terms like 'Briston' and 'cymder' as well as the faded text along with the brake cups which pop up all over the place.

​

I'm starting to think we should consider contributing towards your upgrading your AI subscription from GPT-4o to one of those latest generation AIs(?)

 

[stevevw]

Your post is riddled with errors.

(1) Neither is example (1) within the Mandelbrot set as it is defined by a different iterative function z → z⁷ + c even though it too is a fractal.
By definition the Mandelbrot set is defined by the iterative function z → z² + c, where z and c are complex numbers and c is an element of the set if |z| or mod(z) never becomes larger than a certain number (usually 2 in computer programs) no matter how many iterations are done.

Its still a Mandelbrot set but the value has changed. You can cube, quadriple or 7 times z and you will still get patterns within the Mandelbrot set because your still only allowing sets that fall within the Madelbrot set.

The shapes just changes. So z cubed + C you go from a cardioid shape in the original to a Nephoid when cubed and instead of circles growing off this as in the original you get cardioids growing off it just like in the original. Everything goes up by 1 as you have increased the number by 1 where the cardioids in the original had 1 valley and now they have two.

You can keep doing this with Z to the 4th, 5th, 6th ect and each time the lobes sprouting off it are 1 less than the power its X by. So Z to the power of 5 gives 4 lobes and to X 6 gives 5 lobes ect.

So with Z to the power of 7 as with your example you get 6 lobes and amazingly we get snowflakes. So its not some random set and is still within the Madlebrot set and is still producing the patterns we see in nature such as fractals, the Golden ratio ect but just more complex.

But the point is its not throwing out the Madelbrot set and is still producing the fractals and number sequences that fall within the set. The amazing thing is they also reflect what we see in nature.

(2) Natural fractals such as trees, coastlines, river systems, galaxies (?), flowers and snowflakes do not fall within within the Mandelbrot set nor in any type of mathematical fractal.

Yes they do as explained above and yes galaxies that align with the Golden ratio which is found in the Madelbrot set.

All these pattens are found in the Madelbrot set

Mathematical fractals have perfect self similarity at any scale, have infinite resolution and precision and are 100% deterministic.

Then its strange how a supposed human mathmatical set can coincidently reproduce the patterns of nature.

Natural fractals are effected by chaos caused by external factors such as gravity, temperature, pressure, wind etc which introduces randomness and breaks down self similarity at both large and small scales.

Then its strange how they form patterns like the Mandelbrot set.

(3) The Golden ratio or Phi is not a derivative of the Mandelbrot set.
The Mandelbrot set belongs to the field of complex dynamics and fractals, the Golden Ratio or Phi is about number theory and geometry.

Yes it is.

Hidden numbers and basic mathematics in the Mandelbrot-Set

Hidden numbers and basic mathematics in the Mandelbrot-Set - Fractal.Institute

To the untrained eye, the Mandelbrot-Set might look chaotic and the arrangement of the patterns seem random. But they are the opposite, they are arranged to very precise rules and you can find all kind of mathematics just by observing the Mandelbrot-Set. Basic Calculus: Counting: The tips of...

fractal.institute

 

[sjastro]

Its still a Mandelbrot set but the value has changed. You can cube, quadriple or 7 times z and you will still get patterns within the Mandelbrot set because your still only allowing sets that fall within the Madelbrot set.

The shapes just changes. So z cubed + C you go from a cardioid shape in the original to a Nephoid when cubed and instead of circles growing off this as in the original you get cardioids growing off it just like in the original. Everything goes up by 1 as you have increased the number by 1 where the cardioids in the original had 1 valley and now they have two.

View attachment 358828
You can keep doing this with Z to the 4th, 5th, 6th ect and each time the lobes sprouting off it are 1 less than the power its X by. So Z to the power of 5 gives 4 lobes and to X 6 gives 5 lobes ect.

View attachment 358829

So with Z to the power of 7 as with your example you get 6 lobes and amazingly we get snowflakes. So its not some random set and is still within the Madlebrot set and is still producing the patterns we see in nature such as fractals, the Golden ratio ect but just more complex.

But the point is its not throwing out the Madelbrot set and is still producing the fractals and number sequences that fall within the set. The amazing thing is they also reflect what we see in nature.

View attachment 358827

Yes they do as explained above and yes galaxies that align with the Golden ratio which is found in the Madelbrot set.

View attachment 358832

View attachment 358836

View attachment 358837

All these pattens are found in the Madelbrot set

View attachment 358831


Then its strange how a supposed human mathmatical set can coincidently reproduce the patterns of nature.

Then its strange how they form patterns like the Mandelbrot set.

Yes it is.

Hidden numbers and basic mathematics in the Mandelbrot-Set

Hidden numbers and basic mathematics in the Mandelbrot-Set - Fractal.Institute

To the untrained eye, the Mandelbrot-Set might look chaotic and the arrangement of the patterns seem random. But they are the opposite, they are arranged to very precise rules and you can find all kind of mathematics just by observing the Mandelbrot-Set. Basic Calculus: Counting: The tips of...

fractal.institute

View attachment 358834

What a nonsensical post.

(1) A Mandelbrot set is a unique set defined by a unique iteration function as demonstrated by your images which are not Mandelbrot sets as they have different shapes and higher powers of z contains values of c not found in a Mandelbrot set.

(2) Next time you go down to the beach take your microscope with you; are you going to tell me a microscopic view is going to reveal the same topography one would expect to see if it is like a Mandelbrot set.

(3) Phi is not spelt as pi and in your final link there is a video that shows pi (not phi) can be approximated using the Mandelbrot set which is a very crude method compared to using functions of real numbers.

 

[Hans Blaster]

Don't tell me FORTRAN.

No, I mean Fortran. FORTRAN is awful. Proper Fortran is not formatted for punch cards.

 

[BCP1928]

No, I mean Fortran. FORTRAN is awful. Proper Fortran is not formatted for punch cards.

That's right. ALGOL is the real thing.

 

[sjastro]

No, I mean Fortran. FORTRAN is awful. Proper Fortran is not formatted for punch cards.

I must used Fortran at work then I never used punch cards but had to keep the computer powered up with peddle power.

 

[Hans Blaster]

I must used Fortran at work then I never used punch cards but had to keep the computer powered up with peddle power.

I learned Fortran in college. It was the modern standard. Later I had to regress an learn the "X3J3/90.4, ISO 1539:1980" standard features and I loath fixed column formatting, equivalence statements, computed goto's, and common blocks. I've spent months purging such things from codes.

 

[SelfSim]

I must used Fortran at work then I never used punch cards but had to keep the computer powered up with peddle power.

There was something quite satisfying about punch cards being eaten by a machine. I think they were developed by aliens from the future who occasionally returned to fix 'em when they broke down. They made unreasonable demands of the Uni students using them. They lived behind a mysterious door, with markings denoting the signatures of the advanced technologies they possessed.

 

[stevevw]

What a nonsensical post.

(1) A Mandelbrot set is a unique set defined by a unique iteration function as demonstrated by your images which are not Mandelbrot sets as they have different shapes and higher powers of z contains values of c not found in a Mandelbrot set.

Ok so I thought the value of the Madelbrot sets is contained in the part of the formula that is C = Z n+1. So long as C = Z n+1 does not quickly increase to infinity but stays low and stable its within the Mandelbrot set regardless of whether Z is squared, cubed or quadrupled.

You will find that even within the Madelbrot set as you magnify into the smaller valleys and branches you will discover different shapes to the original shape of the Madelbrot set. You will begin to see different fractals and even sea horse shapes that are not in the unmagnified Mandelbrot shape.

It is the same for when we increase from Zn squared to cubed or quadruple or the power of 6 or 7. You see all the original shapes of the Madelbrot set (more cardioids and increased complex fractals based on the original) but just as the original shape changed with increasing magnification getting more complex so to does the cubed and quadruple examples of the original Mandelbrot set adding new shapes based on the same original formula.

Because they still contain the numbers only allowed within the Madelbrot set. They are still stable numbers that don't explode into big increases to infinity but remain stable within the Madelbrot set. They are expansions of the same formula and not a completely different or random set.

Mandelbrot Magic
Higher Order Exponents
Now that you understand a litlle bit about the functioning of the Mandelbrot Set, we can start to change the equation and explore what happens to the resulting fractal image. Below you will find the Mandlebrot Set and Julia Sets for Z2 + C followed by the Mandelbrot Set and Julia Sets that result from iterating Z4 + C.

Fractal Foundation Online Course - Chapter 1 - FRACTALS IN NATURE

fractalfoundation.org

So why are these increased powers added to the Madelbrot set still called Madelbrot sets ie Mandelbrot Set and Julia Sets that result from iterating Z4 + C. Or in my examples to the power of Z5, Z6 and your example that creates snowflakes Z7. They are still classed as Madelbrots because they still use the sets within the Madelbrot set and don't deviate beyond them.

Higher Order Mandelbrot Sets and their Varying Shapes
It is observed here that as n is increased, the number of cardioids also increases. For n = 2 there is only one cardioid which is the central part of the Mandelbrot set. For n = 3 there are two cardioids connected back-to-back. Similarly, for n = 4 and n = 5 there are three and four cardioids respectively connected back-to-back

https://arxiv.org/pdf/2112.10908

(2) Next time you go down to the beach take your microscope with you; are you going to tell me a microscopic view is going to reveal the same topography one would expect to see if it is like a Mandelbrot set.

I am not sure what you mean. How does this negate that we see the math and geometry of the Madelbrot set in nature. You don't need a microscope. Pick up most shells on the beach and you will see it with the naked eye.

Look a a fern tree which is often found around tropical beaches and you will see the same patterns. The rivers running into the sea have the same patterns and so do the clouds above you when your sunbaking on the beach.

(3) Phi is not spelt as pi and in your final link there is a video that shows pi (not phi) can be approximated using the Mandelbrot set which is a very crude method compared to using functions of real numbers.

I never said they were the same. Phi is the Golden ratio which I have referred to many times. Phi and Pi are two different mathmatical and geometric measures but also related within the overall concept of Golden or Sacred numbers.

The number φ, (Phi) generally known as the Golden Ratio, is simply the smallest of the Golden Numbers. The number π, (Pi) the ratio of the circumference to the diameter of a circle, is related to the largest of the Golden Numbers.

These are all found within the Mandelbrot set and the pre dynastic vases and other works of the Egyptians.

 

[sjastro]

Ok so I thought the value of the Madelbrot sets is contained in the part of the formula that is C = Z n+1. So long as C = Z n+1 does not quickly increase to infinity but stays low and stable its within the Mandelbrot set regardless of whether Z is squared, cubed or quadrupled.

You will find that even within the Madelbrot set as you magnify into the smaller valleys and branches you will discover different shapes to the original shape of the Madelbrot set. You will begin to see different fractals and even sea horse shapes that are not in the unmagnified Mandelbrot shape.

It is the same for when we increase from Zn squared to cubed or quadruple or the power of 6 or 7. You see all the original shapes of the Madelbrot set (more cardioids and increased complex fractals based on the original) but just as the original shape changed with increasing magnification getting more complex so to does the cubed and quadruple examples of the original Mandelbrot set adding new shapes based on the same original formula.

Because they still contain the numbers only allowed within the Madelbrot set. They are still stable numbers that don't explode into big increases to infinity but remain stable within the Madelbrot set. They are expansions of the same formula and not a completely different or random set.

View attachment 358840

Do you even know what a complex number is and please don’t give me a stack of links which doesn’t prove anything.
The modulus of a complex number |z| is the equation of a circle, a point c which is a complex number is part of the Mandelbrot set if after a large number of iterations (do you know what an iteration is) the modulus remains finite and is assigned a radius of 2 in computer programs.

How you can state that fractals of higher powers of z are expansions of the Mandelbrot set is only made by individuals who do not understand simple algebra.

Ever heard of the binomial theorem?

Mandelbrot Magic
Higher Order Exponents
Now that you understand a litlle bit about the functioning of the Mandelbrot Set, we can start to change the equation and explore what happens to the resulting fractal image. Below you will find the Mandlebrot Set and Julia Sets for Z2 + C followed by the Mandelbrot Set and Julia Sets that result from iterating Z4 + C.

Fractal Foundation Online Course - Chapter 1 - FRACTALS IN NATURE

fractalfoundation.org

So why are these increased powers added to the Madelbrot set still called Madelbrot sets ie Mandelbrot Set and Julia Sets that result from iterating Z4 + C. Or in my examples to the power of Z5, Z6 and your example that creates snowflakes Z7. They are still classed as Madelbrots because they still use the sets within the Madelbrot set and don't deviate beyond them.

Higher Order Mandelbrot Sets and their Varying Shapes
It is observed here that as n is increased, the number of cardioids also increases. For n = 2 there is only one cardioid which is the central part of the Mandelbrot set. For n = 3 there are two cardioids connected back-to-back. Similarly, for n = 4 and n = 5 there are three and four cardioids respectively connected back-to-back

https://arxiv.org/pdf/2112.10908

The iteration of higher powered values of z are known as generalized Mandelbrot sets which are not part of the classical Mandelbrot set z →z² +c.
Here is a comparison of the Mandelbrot set with z → z⁷ + c.

​

They are completely different sets, in the interval -2.0 ≤ Real Axis ≥-1.5 are values contained in Mandelbrot set but not in the z → z⁷ + c fractal while the fractal contains values not found in the Mandelbrot set.

Snowflakes are not represented by the z → z⁷ + c fractal since they exhibit a six-fold rotational symmetry due to the molecular arrangement of water molecules in ice, which is related to its hexagonal crystal lattice structures.
Due to the 7th power, the fractal results in a sevenfold rotational symmetry.

I am not sure what you mean. How does this negate that we see the math and geometry of the Madelbrot set in nature. You don't need a microscope. Pick up most shells on the beach and you will see it with the naked eye.

Look a a fern tree which is often found around tropical beaches and you will see the same patterns. The rivers running into the sea have the same patterns and so do the clouds above you when your sunbaking on the beach.

I was referring to complete loss of self similarity at small and large scales for naturally occurring fractals unlike mathematical fractals.

I never said they were the same. Phi is the Golden ratio which I have referred to many times. Phi and Pi are two different mathmatical and geometric measures but also related within the overall concept of Golden or Sacred numbers.

The number φ, (Phi) generally known as the Golden Ratio, is simply the smallest of the Golden Numbers. The number π, (Pi) the ratio of the circumference to the diameter of a circle, is related to the largest of the Golden Numbers.

These are all found within the Mandelbrot set and the pre dynastic vases and other works of the Egyptians.

Then why did you assert that phi is found in the Mandelbrot set by referring to a link containing a description for pi?

 

[stevevw]

Do you even know what a complex number is and please don’t give me a stack of links which doesn’t prove anything.
The modulus of a complex number |z| is the equation of a circle, a point c which is a complex number is part of the Mandelbrot set if after a large number of iterations (do you know what an iteration is) the modulus remains finite and is assigned a radius of 2 in computer programs.

How you can state that fractals of higher powers of z are expansions of the Mandelbrot set is only made by individuals who do not understand simple algebra.

View attachment 358848

Ever heard of the binomial theorem?

I am reasonable at maths. Certainly not an expert. But I understand the basic concept of number sets, iterations and and how that applies to the Madelbrot set. I did do some research lol.

As far as I understand the part I highlighted the (point (c)) is the key. So long as the iterations of the number as you say "after a large number of iterations the modulus remains finite" then it falls within the Madelbrot set. So it doesn't matter if we cube or quadruple z so long as the numbers remain infinite at point c.

So the general Madelbrots are still within the Mandelbrot sets but just powered versions that are producing variations of the same formula at increasing complexity.

Let me ask you a question. Does the patterns produced by the increased powers of z also contain only numbers that fall within the Madelbrot set. That is they do not contain numbers that fall outside and the iterations remain finite.

The iteration of higher powered values of z are known as generalized Mandelbrot sets which are not part of the classical Mandelbrot set z →z² +c.

Yes and I never said that. I said these generalized Mandelbrot sets are still Mandelbrot sets showing even more complexity and still within what we see in nature. Such as the snowflake shapes within the Madelbrot set pattern when we change z to the power of 7. The patterns though powered still contain the same iterations and sequences but now displaying new complexity and patterns and yet still to the same formula.

Here is a comparison of the Mandelbrot set with z → z⁷ + c.

View attachment 358850
​

They are completely different sets, in the interval -2.0 ≤ Real Axis ≥-1.5 are values contained in Mandelbrot set but not in the z → z⁷ + c fractal while the fractal contains values not found in the Mandelbrot set.

Yes and I said that. There will be completely different patterns and shapes. But they all stem from the same formula for determining the set. They still have to contain the numbers that remain finite. So its the same formula creating new patterns. Its really a doubling and tripling ect of the same formula.

If you notice throughout all these examples though patterns change there are some basic shapes and patterns that remain and are what is expanded on. Like the cardioid, the branches, and circles. For example the original Madelbrot pattern is based on a cardioid which two same size circles drawn around each other.

The same with plugging the power of 3 into Z. You get the same basic formula but now its a nephloid where instead of the same size circle we have a circle half the size being drawn around the central circle to form a nephloid. The shapes and patterns are still based on the same formula ( iterations remain finite) but new variations are created.

Notice that instead of small circles around the outside its now small cardioids. It is the same for all the patterns and newly created shapes are incorporated as each level is increased while adding new patterns and shapes based on the same formula.

The snowflake pattern you get with z⁷ + c stems from the same formula. I am not sure what they call this maybe a hexoid. In fact it seems we can keep increasing the power to infinity and the strange thing is we end up back at a circle.

So I would say each increase in power is based on the circle and circles within circles as the basisfor each level of power with all the associated patterns until it becomes a circle again. The point is its the same formula all through each power level which is bringing with it increased complexity based on the same formula (iterations remain infinite.

But I find it facinating that even though the iterations much remain infinite the increase in complexity can go to infinity and when it reaches infinity it reverts back to the simple circle.

But notice no matter how we increase the power the sets still fall within the 'finite iterations on the graph. They conform to a limited formula as opposed to other sets and numbers that fall outside that formula which would cause the patterns and shapes to explode into chaos or some random shape.

Anyway thats how I understood how variations of shapes and patterns in these general Madelbrots still remain within the formula and the formular also contains fractals, Golden rations, Phi, Pi and this is reflected in nature and the Egyptian vases and other works.

It seems to me that these shapes and patterns could be known even by ancient people who lacked modern tech. In some ways you could say that these shapes and patterns may be a form of advanced tech but in a different way to how we understand things.

Snowflakes are not represented by the z → z⁷ + c fractal since they exhibit a six-fold rotational symmetry due to the molecular arrangement of water molecules in ice, which is related to its hexagonal crystal lattice structures.
Due to the 7th power, the fractal results in a sevenfold rotational symmetry.

But the power of 7 actually creates 6 buds and thus a snowflake shape. The buds created are always 1 less than the power used. So quadrupling the power of z actually creates a 3 sided shape.

I was referring to complete loss of self similarity at small and large scales for naturally occurring fractals unlike mathematical fractals.

You mean like the Barnesley Fern a mathmatical pattern where each small leaf is a minature version of the whole thing.

But would not fractals within the Madelbrot set still have the same formula thus stem from the same basis regardless of size and patterns. They may become m,ore complex, trippling quadrupling in branches and expansions of the same basic shapes and patterns just becoming more complex.

Then why did you assert that phi is found in the Mandelbrot set by referring to a link containing a description for pi?

Oh I see what you mean. I didn't even notice it nor understood what you were saying until you clarified this now. OK so I must have slipped in my thinking and got Phi and Pi mixed up in my rush or distraction or whatever it was lol.

But then it doesn't really matter if I mixed them up because they are both in the Madelbrot set.

 

[SelfSim]

You mean like the Barnesley Fern a mathmatical pattern where each small leaf is a minature version of the whole thing.
View attachment 358858

The Barnsley Fern functions were created by Barnsley in order to resemble the black spleenwort, Asplenium adiantum-nigrum .. and not the other way around.
His fern-like pattern consists of four, separate iterated functions acting together in a computer simulated environment.
He says:

IFSs, {Iterated Function Systems}, provide models for certain plants, leaves, and ferns, by virtue of the self-similarity which often occurs in branching structures in nature. But nature also exhibits randomness and variation from one level to the next; no two ferns are exactly alike, and the branching fronds become leaves at a smaller scale. V-variable fractals allow for such randomness and variability across scales, while at the same time admitting a continuous dependence on parameters which facilitates geometrical modelling. These factors allow us to make the hybrid biological models... ...we speculate that when a V -variable geometrical fractal model is found that has a good match to the geometry of a given plant, then there is a specific relationship between these code trees and the information stored in the genes of the plant.

Thus, the models he created contribute towards understanding how DNA information is replicated, transcribed and transformed.

I had a rainforest fern which looked very much like his simulated model. It had visible 'defects' in the pattern regularity, visible to the naked eye. I concluded that Barnsley's fern, was not my fern, (nor was it the product of some computer generated fractal). Its DNA containing seeds originated from a rainforest environment.

 

[sjastro]

I am reasonable at maths. Certainly not an expert. But I understand the basic concept of number sets, iterations and and how that applies to the Madelbrot set. I did do some research lol.

As far as I understand the part I highlighted the (point (c)) is the key. So long as the iterations of the number as you say "after a large number of iterations the modulus remains finite" then it falls within the Madelbrot set. So it doesn't matter if we cube or quadruple z so long as the numbers remain infinite at point c.

So the general Madelbrots are still within the Mandelbrot sets but just powered versions that are producing variations of the same formula at increasing complexity.

Let me ask you a question. Does the patterns produced by the increased powers of z also contain only numbers that fall within the Madelbrot set. That is they do not contain numbers that fall outside and the iterations remain finite.

Yes and I never said that. I said these generalized Mandelbrot sets are still Mandelbrot sets showing even more complexity and still within what we see in nature. Such as the snowflake shapes within the Madelbrot set pattern when we change z to the power of 7. The patterns though powered still contain the same iterations and sequences but now displaying new complexity and patterns and yet still to the same formula.

Yes and I said that. There will be completely different patterns and shapes. But they all stem from the same formula for determining the set. They still have to contain the numbers that remain finite. So its the same formula creating new patterns. Its really a doubling and tripling ect of the same formula.

If you notice throughout all these examples though patterns change there are some basic shapes and patterns that remain and are what is expanded on. Like the cardioid, the branches, and circles. For example the original Madelbrot pattern is based on a cardioid which two same size circles drawn around each other.
View attachment 358852

The same with plugging the power of 3 into Z. You get the same basic formula but now its a nephloid where instead of the same size circle we have a circle half the size being drawn around the central circle to form a nephloid. The shapes and patterns are still based on the same formula ( iterations remain finite) but new variations are created.

Notice that instead of small circles around the outside its now small cardioids. It is the same for all the patterns and newly created shapes are incorporated as each level is increased while adding new patterns and shapes based on the same formula.

View attachment 358853

The snowflake pattern you get with z⁷ + c stems from the same formula. I am not sure what they call this maybe a hexoid. In fact it seems we can keep increasing the power to infinity and the strange thing is we end up back at a circle.

So I would say each increase in power is based on the circle and circles within circles as the basisfor each level of power with all the associated patterns until it becomes a circle again. The point is its the same formula all through each power level which is bringing with it increased complexity based on the same formula (iterations remain infinite.

But I find it facinating that even though the iterations much remain infinite the increase in complexity can go to infinity and when it reaches infinity it reverts back to the simple circle.


View attachment 358854

View attachment 358855

But notice no matter how we increase the power the sets still fall within the 'finite iterations on the graph. They conform to a limited formula as opposed to other sets and numbers that fall outside that formula which would cause the patterns and shapes to explode into chaos or some random shape.

Anyway thats how I understood how variations of shapes and patterns in these general Madelbrots still remain within the formula and the formular also contains fractals, Golden rations, Phi, Pi and this is reflected in nature and the Egyptian vases and other works.

It seems to me that these shapes and patterns could be known even by ancient people who lacked modern tech. In some ways you could say that these shapes and patterns may be a form of advanced tech but in a different way to how we understand things.

But the power of 7 actually creates 6 buds and thus a snowflake shape. The buds created are always 1 less than the power used. So quadrupling the power of z actually creates a 3 sided shape.

You mean like the Barnesley Fern a mathmatical pattern where each small leaf is a minature version of the whole thing.
View attachment 358858

But would not fractals within the Madelbrot set still have the same formula thus stem from the same basis regardless of size and patterns. They may become m,ore complex, trippling quadrupling in branches and expansions of the same basic shapes and patterns just becoming more complex.

Oh I see what you mean. I didn't even notice it nor understood what you were saying until you clarified this now. OK so I must have slipped in my thinking and got Phi and Pi mixed up in my rush or distraction or whatever it was lol.

But then it doesn't really matter if I mixed them up because they are both in the Madelbrot set.

For someone who claims to be reasonable at mathematics and then provides example after example you don’t know what you are talking about is classic Dunning Kruger behaviour.
All you have demonstrated is how to use the bold key without understanding the emboldened statement.

A case in point is the nonsense of associating the emboldened statement with some general or same formula and failing to realise it is the function being iterated which determines if any given value of c remains in what is termed a stable orbit for the given modulus of the iterated function.

The fractals clearly show similarities in fact are identical in the red regions corresponding to unstable orbits rapidly diverging to infinity after a few iterations, but it is the stable orbits in black which are the regions of interest.
The common denominator in the black regions is the number of rotational symmetries increases by one as the power of the iterated function z increases by one and not due to some general or same formula.

This also applies to the question of circles which generate the cardioid, nephroid, deltoid and higher cusped hypocycloids where the z power of the general Mandelbrot set is greater than four.

These hypocycloids are defined by the number of cusps in the central region of the set and correspond to critical points where the derivative of the set is zero.

For the generalized Mandelbrot set:

The solution indicates there are n-1 critical points leading to the characteristic n-1 cusps which define the hypocycloid.
When n → ∞ these critical points become extremely dense and the set forms a circle at the limit.
Note once again it depends on the power of the generalized Mandelbrot set not some general or same formula.

Another example is claiming generalized Mandelbrot set accurately describes snowflakes when the Koch fractal best describes these.

You are carrying considerable emotional baggage in this thread and have adopted a contrarian approach as a matter of principle.
I suggest you go to a maths forum to get an education on Mandelbrot and generalized sets followed by a trip to a physics forum where they will explain why naturally occurring fractals are poor examples of mathematical fractals.

You might actually learn something because you would have no reason for being argumentative as you are on this site.

 

[stevevw]

The Barnsley Fern functions were created by Barnsley in order to resemble the black spleenwort, Asplenium adiantum-nigrum .. and not the other way around.
His fern-like pattern consists of four, separate iterated functions acting together in a computer simulated environment.
He says:

Thus, the models he created contribute towards understanding how DNA information is replicated, transcribed and transformed.

I had a rainforest fern which looked very much like his simulated model. It had visible 'defects' in the pattern regularity, visible to the naked eye. I concluded that Barnsley's fern, was not my fern, (nor was it the product of some computer generated fractal). Its DNA containing seeds originated from a rainforest environment.

We have plenty Barnsley type ferns in Australia.

 

[SelfSim]

We have plenty Barnsley type ferns in Australia.
View attachment 358885

Incorrect.

 

[stevevw]

For someone who claims to be reasonable at mathematics and then provides example after example you don’t know what you are talking about is classic Dunning Kruger behaviour.
All you have demonstrated is how to use the bold key without understanding the emboldened statement.

Thats a bit unfair. You would not make a good teacher. Imagine talking to students like that.

A case in point is the nonsense of associating the emboldened statement with some general or same formula and failing to realise it is the function being iterated which determines if any given value of c remains in what is termed a stable orbit for the given modulus of the iterated function.

Why be so pedantic. What I meant by formula was the same thing. The point was that each stable orbit still falls within the Madelbrot set. It was you who claimed the generalised Mandelbrots were not Mandelbrots.

So did humans just happen to fluke creating the Madelbrot reflecting nature or was it something they discovered and the Madelbrot set is just a tool humans created to discover these patterns that already existed in nature.

View attachment 358867​

The fractals clearly show similarities in fact are identical in the red regions corresponding to unstable orbits rapidly diverging to infinity after a few iterations, but it is the stable orbits in black which are the regions of interest.
The common denominator in the black regions is the number of rotational symmetries increases by one as the power of the iterated function z increases by one and not due to some general or same formula.

This also applies to the question of circles which generate the cardioid, nephroid, deltoid and higher cusped hypocycloids where the z power of the general Mandelbrot set is greater than four.

Ah so thats their names. I could not find the names of the general Madelbrots when increased in power.

These hypocycloids are defined by the number of cusps in the central region of the set and correspond to critical points where the derivative of the set is zero.

For the generalized Mandelbrot set:

View attachment 358880

The solution indicates there are n-1 critical points leading to the characteristic n-1 cusps which define the hypocycloid.
When n → ∞ these critical points become extremely dense and the set forms a circle at the limit.
Note once again it depends on the power of the generalized Mandelbrot set not some general or same formula.

Another example is claiming generalized Mandelbrot set accurately describes snowflakes when the Koch fractal best describes these.

You are carrying considerable emotional baggage in this thread and have adopted a contrarian approach as a matter of principle.

No I have provided factual evidence and you have not addressed it. Pointing out the signatures in stones don't match the tools in the archeological records is not arguementative, not contrary in any unjustified way. Its a simple and obvious factual observation. Its you who are being arguementative and contrary in making all sorts of logical fallacies to deny this truth.

I suggest you go to a maths forum to get an education on Mandelbrot and generalized sets followed by a trip to a physics forum where they will explain why naturally occurring fractals are poor examples of mathematical fractals.

I don't need to do anything as its all irrelevant. All I did was point out that the Madelbrot set is reflected in nature and that the Golden ration within the Madelbrot set is also found in the ancient Egyptian vases and other works.

I don't need to be an expert in math to know this. Your trying kill this truth with irrelevant math lessons that don't change this fact. But I appreciate your effort and I do learn all the time. But it was not necessary for my point.

You might actually learn something because you would have no reason for being argumentative as you are on this site.

Don't mistake my views as arguements. Perhaps thats your probloem. It is not I who is taking an arguementative position. When you start posts you rhetoric like "you know nothing' and "are you stupid" your actually being arguementive.

By the way you have been wrong on many occassions. But I don't deride you, don't call you dumb. I just accept that this is your point of view. Everyone needs to learn but you don't teach people by calling them names. That actually turns them off. You would not make a good teacher thats for sure.

 

[stevevw]

Incorrect.

What, Australia has over 400 species of ferns and you don't think we can find Barnsley type ferns amoung them.